We have the following question :
If $D$ denotes the unit ball in $\mathbb C$, then let $f : \bar D \to \mathbb C$ such that $f$ is continuous in $\bar D$ and analytic on $D$. If $f(e^{it}) = 0$ for all $0 < t < \frac \pi 2$ then $f \equiv 0$.
Yes, the idea must be maximum modulus theorem. But I don't see a candidate function : I think we must multiply $f$ with some holomorphic non-zero function so that the newly produced function $g$ is identically zero on the unit circle. But I am unable to cook up such a function to multiply with $f$.
I don't see any further tricks to this : the setting of being analytic on the interior and continuous on the closure is maximum modulus fodder, I would presume.